We prove new upper bounds for the number of representations of an arbitrary rational number as a sum of three unit fractions. In particular, for fixed m there are at most ${\cal O}_{\epsilon }(n^{{3}/{5}+\epsilon })$ solutions of ${m}/{n} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$. This improves upon a result of Browning and Elsholtz (2011) and extends a result of Elsholtz and Tao (2013) who proved this when m=4 and n is a prime. Moreover, there exists an algorithm finding all solutions in expected running time ${\cal O}_{\epsilon }(n^{\epsilon }({n^3}/{m^2})^{{1}/{5}})$, for any $\epsilon \gt 0$. We also improve a bound on the maximum number of representations of a rational number as a sum of k unit fractions. Furthermore, we also improve lower bounds. In particular, we prove that for given $m \in {\open N}$ in every reduced residue class e mod f there exist infinitely many primes p such that the number of solutions of the equation ${m}/{p} = {1}/{a_1} + {1}/{a_2} + {1}/{a_3}$ is $\gg _{f,m} \exp (({5\log 2}/({12\,{\rm lcm} (m,f)}) + o_{f,m}(1)) {\log p}/{\log \log p})$. Previously, the best known lower bound of this type was of order $(\log p)^{0.549}$.

In this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality:

$$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$

${\cal D}_0^{2,2} $

This paper is concerned with the travelling waves for a class of non-local dispersal non-cooperative system, which can model the prey-predator and disease-transmission mechanism. By the Schauder's fixed-point theorem, we first establish the existence of travelling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations, whose bounds are deduced from a precise analysis. Further, we characterize the minimal wave speed of travelling waves and obtain the non-existence of travelling waves with slow speed. Finally, we apply the general results to an epidemic model with bilinear incidence for its propagation dynamics.

This paper is concerned with the fractional system

\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u(x) = \vert x \vert ^a v^p(x), &x\in\mathbb{R}^n_+,\\ (-\Delta)^{\frac{\beta}{2}} v(x) = \vert x \vert ^b u^q(x), &x\in\mathbb{R}^n_+,\\ u(x)=v(x)=0, &x\in\mathbb{R}^n{\setminus}\mathbb{R}^n_+, \end{cases}

$p \leqslant \frac {n+\alpha +2a}{n-\beta }$

$q \leqslant \frac {n+\beta +2b}{n-\alpha }$

$p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$

We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.

In this paper, we consider the nonlinear Choquard equation

$$-\Delta u + V(x)u = \left( {\int_{{\open R}^N} {\displaystyle{{G(u)} \over { \vert x-y \vert ^\mu }}} \,{\rm d}y} \right)g(u)\quad {\rm in}\;{\open R}^N, $$

$G(u)=\int ^u_0g(s)\,{\rm d}s$

In this paper, we study the existence of weak solutions of the quasilinear equation

\begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}

We consider a second-order elliptic operator L in skew product of an ordinary differential operator L1 on an interval (a, b) and an elliptic operator on a domain D2 of a Riemannian manifold such that the associated heat kernel is intrinsically ultracontractive. We give criteria for criticality and subcriticality of L in terms of a positive solution having minimal growth at η (η = a, b) to an associated ordinary differential equation. In the subcritical case, we explicitly determine the Martin compactification and Martin kernel for L on the basis of [24]; in particular, the Martin boundary over η is either one point or a compactification of D2, which depends on whether an associated integral near η diverges or converges. From this structure theorem we show a monotonicity property that the Martin boundary over η does not become smaller as the potential term of L1 becomes larger near η.

Let X ⊂ ℂn be an equidimensional complex algebraic set and let f: X → ℂ be a polynomial function. For each c ∈ ℂ, we define the global Brasselet number of f at c, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of f at c. Then we establish several formulas relating these numbers to the topology of X and the critical points of f.

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely

$$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$

We consider weak solutions $u:\Omega _T\to {\open R}^N$ to parabolic systems of the type

$$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$

$x\mapsto a(x,t,\xi )$

$D_xa(\cdot ,\cdot ,\xi )$

$L^\alpha (0,T;L^\beta (\Omega ))$

$\alpha ,\beta $

$$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$

$$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$

$u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

We study the non-autonomously forced Burgers equation

$$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$

We apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$

$n\ges 222$

$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

In this article, the existence of heteroclinic solution of a class of generalized Hamiltonian system with potential $V : {\open R}^{n} \longmapsto {\open R}$ having a finite or infinite number of global minima is studied. Examples include systems involving the p-Laplacian operator, the curvature operator and the relativistic operator. Generalized conservation of energy is established, which leads to the property of equipartition of energy enjoyed by heteroclinic solutions. The existence problem of heteroclinic solution is studied using both variational method and the metric method. The variational approach is classical, while the metric method represents a more geometrical point of view where the existence problem of heteroclinic solution is reduced to that of geodesic in a proper length metric space. Regularities of the heteroclinic solutions are discussed. The results here not only provide alternative solution methods for Φ-Laplacian systems, but also improve existing results for the classical Hamiltonian system. In particular, the conditions imposed upon the potential are very mild and new proof for the compactness is given. Finally in ℝ2, heteroclinic solutions are explicitly written down in closed form by using complex function theory.

We prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of ℝn space. In particular, the result applies also to some operators in Carnot groups.

In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspace, each network admissible system restricted to that subspace is a dynamical system consistent with a smaller network called a quotient network. We characterize networks such that: the network is a subnetwork of its fundamental network, and the network is a fundamental network. Moreover, we prove that the fundamental network construction preserves the quotient relation and it transforms the subnetwork relation into the quotient relation. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.